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Journal of Physics A Mathematical and Theoretical

Quasi-particle approach for lattice Hamiltonians with large coordination numbers

Patrick Navez, Friedemann Queißer, Ralf Schützhold

Published May 19, 2014• 18 citations
Publisher

Abstract

In many condensed-matter systems, it is very useful to introduce a quasi-particle approach, which is based on some sort of linearization around a suitable background state. In order to be a systematic and controlled approximation, this linearization should be justified by an expansion into the powers of some small control parameter. Here, we present a method for general lattice Hamiltonians with large coordination numbers Z 1, which is based on an expansion into the powers of 1/Z. In order to demonstrate the generality of our method, we apply it to various spin systems, as well as the Bose and Fermi–Hubbard model.

DOI
10.1088/1751-8113/47/22/225004

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Quasi-particle approach for general lattice Hamiltonians

Patrick Navez, Friedemann Queißer, Ralf Schützhold

In many condensed-matter systems, it is very useful to introduce a quasi-particle approach, which is based on some sort of linearization around a suitable background state. In order to be a systematic and controlled approximation, this linearization should be justified by an expansion into powers of some small control parameter. Here, we present a method for general lattice Hamiltonians with large coordination numbers Z >> 1, which is based on an expansion into powers of 1/Z. In order to demonstrate the generality of our method, we apply it to various spin systems, as well as the Bose and Fermi Hubbard model.

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Comment on the paper "Quasi-particle approach for lattice Hamiltonians with large coordination numbers" by P. Navez, F. Queisser and R. Schützhold - J. Phys. A: Math. Theor. 47 225004 (2014)

Demetra Psiachos

This comment regards a central aspect of the referred-to paper, the issue of convergence of the large coordination-number expansion. Perturbation expansions of expressions containing a large number of parameters are generally invalid due to the non-analyticity of the expanded expressions. I refer to recent work where these issues are analyzed and discussed in detail in relation to a benchmark example of a cluster model. As discussed therein, methods which are uncontrollable and for which their convergence is not foreseeable are not only useless but can mislead, particularly if models derived from them are used to interpret experiments.

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Correlations in the Bose & Fermi Hubbard Model

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We study the Bose-Hubbard and Fermi-Hubbard model in the limit of large coordination numbers Z (i.e., many tunnelling partners). Via a controlled expansion into powers of 1/Z, we establish a hierarchy of correlations, which facilitates an approximate analytic solution of the quantum evolution. For the Bose-Hubbard model, we derive the growth of phase coherence after a quench from the Mott to the superfluid phase. For a quench within the Mott phase, we find that various local observables approach a quasi-equilibrium state after a finite period of time. However, this state is not thermal, i.e., real thermalisation -- if it occurs -- requires much longer time scales. For a tilted lattice in the Mott state, we calculate the tunnelling probability and find a remarkable analogy to the Sauter-Schwinger effect (i.e., electron-positron pair creation out of the vacuum due to a strong electric field). These analytical results are compared to numerical simulations for finite lattices in one and two dimensions and we find qualitative agreement. Finally, we generalize these studies to the more involved case of the Fermi-Hubbard model.

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Perturbation series expansion methods are sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. This book gives a comprehensive guide to the use of series expansion methods for investigating phase transitions and critical phenomena, and lattice models of quantum magnetism, strongly correlated electron systems and elementary particles. Early chapters cover the classical treatment of critical phenomena through high-temperature expansions, and introduce graph theoretical and combinatorial algorithms. The book then discusses high-order linked-cluster perturbation expansions for quantum lattice models, finite temperature expansions, and lattice gauge models. Also included are numerous detailed examples and case studies, and an accompanying resources website, www.cambridge.org/9780521842426, contains programs for implementing these powerful numerical techniques. A valuable resource for graduate students and postdoctoral researchers working in condensed matter and particle physics, this book will also be useful as a reference for specialized graduate courses on series expansion methods.

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In many condensed-matter systems, it is very useful to introduce a quasi-particle approach, which is based on some sort of linearization around a suitable background state. In order to be a systematic and controlled approximation, this linearization should be justified by an expansion into the powers of some small control parameter. Here, we present a method for general lattice Hamiltonians with large coordination numbers Z 1, which is based on an expansion into the powers of 1/Z. In order to demonstrate the generality of our method, we apply it to various spin systems, as well as the Bose and Fermi–Hubbard model.
In many condensed-matter systems, it is very useful to introduce a quasi-particle approach, which is based on some sort of linearization around a suitable background state. In order to be a systematic and controlled approximation, this linearization should be justified by an expansion into powers of some small control parameter. Here, we present a method for general lattice Hamiltonians with large coordination numbers Z >> 1, which is based on an expansion into powers of 1/Z. In order to demonstrate the generality of our method, we apply it to various spin systems, as well as the Bose and Fermi Hubbard model.
This comment regards a central aspect of the referred-to paper, the issue of convergence of the large coordination-number expansion. Perturbation expansions of expressions containing a large number of parameters are generally invalid due to the non-analyticity of the expanded expressions. I refer to recent work where these issues are analyzed and discussed in detail in relation to a benchmark example of a cluster model. As discussed therein, methods which are uncontrollable and for which their convergence is not foreseeable are not only useless but can mislead, particularly if models derived from them are used to interpret experiments.
We study the Bose-Hubbard and Fermi-Hubbard model in the limit of large coordination numbers Z (i.e., many tunnelling partners). Via a controlled expansion into powers of 1/Z, we establish a hierarchy of correlations, which facilitates an approximate analytic solution of the quantum evolution. For the Bose-Hubbard model, we derive the growth of phase coherence after a quench from the Mott to the superfluid phase. For a quench within the Mott phase, we find that various local observables approach a quasi-equilibrium state after a finite period of time. However, this state is not thermal, i.e., real thermalisation -- if it occurs -- requires much longer time scales. For a tilted lattice in the Mott state, we calculate the tunnelling probability and find a remarkable analogy to the Sauter-Schwinger effect (i.e., electron-positron pair creation out of the vacuum due to a strong electric field). These analytical results are compared to numerical simulations for finite lattices in one and two dimensions and we find qualitative agreement. Finally, we generalize these studies to the more involved case of the Fermi-Hubbard model.
We review recent advances in the calculations of high-order convergent expansions for quantum many-particle systems. Calculations for ground state properties, including correlation functions and static susceptibilities, for spin models as well as for models of many fermions, such as Hubbard and Kondo models, are discussed. A historical perspective to the subject is provided. Recently important technical advances have been made in perturbative calculations of the excitation spectra of quantum many-particle systems, which enable the calculation of these spectra to high orders. The method, along with its applications, are explained. Fairly comprehensive, though simplified, algorithms for generating lists of relevant clusters, their lattice embeddings and subclusters are presented. The perturbative recursion relations and their computer implementation are also discussed in detail. A compilation is made of various series expansion studies that have been carried out for condensed matter problems. The scope and limitations of these methods are explained, and several open problems are noted.
Perturbation series expansion methods are sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. This book gives a comprehensive guide to the use of series expansion methods for investigating phase transitions and critical phenomena, and lattice models of quantum magnetism, strongly correlated electron systems and elementary particles. Early chapters cover the classical treatment of critical phenomena through high-temperature expansions, and introduce graph theoretical and combinatorial algorithms. The book then discusses high-order linked-cluster perturbation expansions for quantum lattice models, finite temperature expansions, and lattice gauge models. Also included are numerous detailed examples and case studies, and an accompanying resources website, www.cambridge.org/9780521842426, contains programs for implementing these powerful numerical techniques. A valuable resource for graduate students and postdoctoral researchers working in condensed matter and particle physics, this book will also be useful as a reference for specialized graduate courses on series expansion methods.